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| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Induction | Obsidian::STEM | algebra::sequence proof |
|
Overview
%%ANKI Cloze The {base case} is to induction whereas {initial conditions} are to recursive definitions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze The {inductive case} is to induction whereas {recurrence relations} are to recursive definitions. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What standard names are given to the cases in an induction proof? Back: The base case and inductive case. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Let (a_n)_{n \geq 0} = P(n) and P(n) \Leftrightarrow n \geq 2. How is (a_n) written with terms expanded?
Back: F, F, T, T, T, \ldots
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
If proving P(n) by weak induction, what are the first five terms of the underlying sequence?
Back: P(0), P(1), P(2), P(3), P(4), \ldots
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What proposition is typically proven in the base case of an inductive proof?
Back: P(n_0) for some n_0 \geq 0.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What proposition is typically proven in the inductive case of an inductive proof?
Back: P(k) \Rightarrow P(k + 1) for all k \geq n_0.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
In weak induction, what special name is given to the antecedent of P(k) \Rightarrow P(k + 1)?
Back: The inductive hypothesis.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze {Closed formulas} are to recursive definitions as {direct proofs} are to proof strategies. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Cloze {Recurrence relations} are to recursive definitions as {induction} is to proof strategies. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What proof strategy is most directly tied to recursion? Back: Induction. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Using typical identifiers, what is the inductive hypothesis of P(n) using weak induction?
Back: Assume P(k) for some k \geq n_0.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
Using typical identifiers, what is the inductive hypothesis of P(n) using strong induction?
Back: Assume P(k) for all k < n.
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic Why is strong induction considered stronger than weak induction? Back: It can be used to solve at least the same set of problems weak induction can. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI
Basic
What contradiction is introduced to explain why the strong induction assumption is valid?
Back: If P(n) is not true for all n, there exists a first n_0 for which \neg P(n_0).
Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
END%%
%%ANKI Basic What distinguishes the base case of weak and strong induction proofs? Back: The latter may have more than one base case. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
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Bibliography
- Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.